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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeJul 13th 2013
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 13th 2013

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJul 13th 2013

Added another example, and a reference.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMar 5th 2017

At duoidal category I have:

• corrected the example of endofunctors (they are not normal in general) and added an example of profunctors.
• generalized the notion of “commutative monoid” to the case of a non-normal duoidal category, in which case the object first has to be assumed to be “strong”.
• mentioned the notion of “virtual duoidal category”, to which it seems nearly all the definitions in a duoidal category can be generalized.
• CommentRowNumber5.
• CommentAuthorJon Beardsley
• CommentTimeSep 1st 2021

I was looking at this entry recently and had a question about it which has a possibly obvious answer: it is mentioned that if A and B are monoidal then Fun(A,B) is equipped with a duoidal structure with one tensor product being the pointwise one, and the other being Day convolution. Later in the entry, it states that the category of monoids with respect to one monoidal structure, in a duoidal category, is monoidal with respect to the other monoidal structure. My question is the following: suppose we take B to be a monoidal category and A=1, the unit category (with respect to the cartesian product of categories). Then algebras in Fun(1,B), with respect to the pointwise monoidal structure are exactly algebras in B, and algebras in Fun(1,B) with respect to the Day convolution are also algebras in B. Then it seems that for Alg(Fun(1,B)), with respect to either monoidal structure, has a monoidal structure given by the underlying monoidal structure in B. However, at least in the usual cases, this cannot happen in a monoidal category, as far as I know.

So, for this to make sense, do we need to require B to be (at least) braided monoidal?